Chap. 7 Relations: The Second Time Around

1. Chap. 7 Relations: The Second Time Around

2. Binary Relation For sets A, B, any subset of A╳Bis called a (binary) relation from A to B. Any subset of A╳A is called a (binary) relation on A.

3. Reflexive Relation • e.g. Given a finite set A with |A|=n. Then, • The number of relations on A is . • 2. The number of reflexive relations on A is .

4. Symmetric Relation e.g. Given a finite set A with |A|=n. Then, 1. The number of symmetic relations on A is . 2. The number of reflexive and symmetic relations on A is .

5. Transitive Relation Let A={1, 2, 3, 4}. Which of the following relation is transitive? a) R1={(1,1), (2,3), (3,4), (2,4)}. b) R2={(1,3), (3,2)}. O X because (1,3), (3,2)∈R2 but (1,3)∉R2 .

6. Antisymmetric Relation Let A={1, 2, 3}. Which of the following relation is antisymmetric? a) R1={={(1,1), (2,2)}. b) R2={(1,2), (2,1), (2,3)}. O X because (1,2), (2,1)∈R2 but 1≠2.

7. Partial Ordering Relation Which of the following relation is a partial order? a) The relation R on the set Z is defined by aRb, or (a, b)∈R, if a≤b. b) Let n∈Z+, For x,y ∈Z, the modulo relation R is defined by xRy if x-y is a multiple of n. c) The relation R on the set A={1,2,3,4} is defined byaRb if a|b. O Total Order X because it is not antisymmetric. O

8. Example 7.15 Let A={1, 2, 4, 8, 16}, the set of positive integer divisors of 16. Define the relation R on the set A by xRy if x divides y. Then, the order pairs from A╳A that comprise R: R= {(1,1), (1,2), (1,4), (1,8), (1,16), (2,2), (2,4), (2,8), (2,16), (4,4), (4,8), (4,16), (8,8), (8,16), (16,16)}.

9. Example 7.15 (2) 1. (c,d)∈R⇔ and , Where m, p∊N with 0≤m≤p≤4. 2. Each possibility for m, p is simply a selection of size 2 from a set of size 5, the set {0,1,2,3,4}, where the repetitions are allowed. 3. Thus, the number of ways to choose m, p is 5. Therefore, the number of order pairs in R is 15.

10. Example 7.15 (3) Let A={1, 2, 3, 4, 6, 12}, the set of positive integer divisors of 12. Define the relation R on the set A by xRy if x divides y. Then, the order pairs from A╳A that comprise R:

11. Example 7.15 (4) 1. (c,d)∈R⇔ where 3. Thus, the number of ways to choose m, p is 4. Similarly, the number of ways to choose n, q is 5. Therefore, the number of order pairs in R is

12. Equivalence Relation • Let A={1, 2, 3}. Which of the following is a equivalence relation? O O O O O

13. Equivalence Relation 2. Equivalence Class

14. Directed Graph V: vertex set E: edge set V: set of vertices E: subset of V╳V

15. Relation and Directed Graph

16. Poset Let

17. Hasse Diagram

18. Hasse Diagram (2) e.g.

19. Total Order Which of the following relation is a total order? O X O . . . . . .

20. Maximal and Minimal Elements

21. Theorem 7.3 • 1. • 2. • 3. • 4.

22. Least and Greatest Elements Which of the following partial orders has a least element and a greatest element ? O O X X

23. Theorem 7.4 1. 2. It suffices to show 3. 4. 5. 6. x=y

24. Partition Let . Which of the following determines a partition of A ? O O O

25. Equivalence Class

26. Theorem 7.6 • 1. It suffices to show . • 2. This is clearly true because . • b) (⇒) 1. It suffices to show • 2. To show , we need to show for all , • . • 3. Clearly, . • 4. Thus, .

27. Theorem 7.6 (2) • 5. To show , we need to show for all • . • 6. • 7. • b) (⇐) 1. • 2.

28. Theorem 7.6 (3) c) 1. 2. 3. 4. 5. 6. 7.

29. Theorem 7.7 1. 2. 3. (x,x)∊R ⇒ 4. (x,y)∊R ⇒ 5. (x,y)∊R and (y,z)∊R ⇒ R is reflexive. x and y are in the cell of the partition ⇒ (y,x)∊R ⇒ R is symmetric. x, y, and z are in the cell of the partition ⇒ (x,z)∊R ⇒ R is transitive.

30. Example 7.59 1. 2. 3. 4.

31. Example 7.59 (2) 1. 2.